#348664

order	512
nilpotency class	3
isoclinism	isoclinic class of #348654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3529	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(18,24)(19,29)(20,33)(21,22)(26,34)(27,28)(31,32)(35,36)(37,40)(39,47)(43,45)(44,48);(17,18,22,27,23,28,21,24)(19,25,29,40,32,41,31,37)(20,34,36,38,35,26,33,42)(30,44,43,47,46,39,45,48);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(17,19)(18,25)(20,30)(21,31)(22,29)(23,32)(24,37)(26,39)(27,40)(28,41)(33,45)(34,44)(35,46)(36,43)(38,47)(42,48);(4,14)(7,12)(9,16)(10,13)(17,20)(18,26)(19,30)(21,33)(22,36)(23,35)(24,38)(25,39)(27,42)(28,34)(29,43)(31,45)(32,46)(37,47)(40,48)(41,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,d^b=dh,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255860);